Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
We don't have any stats on how common this name is. This is probably because it's very rare in the UK.
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The Croix surname in historical dictionaries
Patronymica Britannica (1860)
DE ST. CROIX. INIany places in France and particularly in Normandy are dedicated to the Holy Cross. The English family left Normandy at the Rev. of the Edict of Nantes, and settled in the island of Jersey from whence they have subsequently transferred themselves to this country.